Measurement and Computation of Streamflow: Volume 2 Computation of Discharge Paper 2175 1982

Measurement and Computation
of Streamflow: Volume 2.
Computation of Discharge
By S. E. RANTZ and others
GEOLOGICAL SURVEY WATER-SUPPLY PAPER 2175
UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1982

UNITED STATES DEPARTMENT OF THE INTERIOR
JAMES G. WATT, Secretary
GEOLOGICAL SURVEY
Dallas L. Peck, Director
First printing 1982
Second printing 1983
Library of Congress Cataloging in Publication Data
Measurement and computation of streamflow.
(Geological Survey Water-Supply Paper 2175)
Includes bibliographies.
Contents: v. 1. Measurement of stage and discharge.-
v. 2. Computation of discharge. 1. Stream
measurements.
I. Rantz, S. E. (Saul Edward), 1911- . II. Series.
TC175.M42 551.48’3’0287 81-607309
AACR2
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C. 20402

CONTENTS
IArt~cle headmgs are lwted in the table of contents only m the volume of the manual m which they occur, but all
chapter titles for the two volumes are llsted m each volume A complete Index covenng both volumes of the
manual appears m each volume 1
VOLUME 1. MEASUREMENT OF STAGE AND DISCHARGE
Chapter Page
1 Introduction ----_----------------------------------~-~-------- 1
2 Selection of Gaging-Station Sites ________________________________ 4
3 Gaging-Station Controls ----------_-_-_-_-- _____________________ 10
4 MeasurementofStage--_-_-__-_______________________________-- 22
5 Measurement of Discharge by Conventional
Current-Meter Method ------------------_- __________________ 79
6 Measurement of Discharge by the Moving-Boat Method __________ 183
7 Measurement of Discharge by Tracer Dilution - _______ .___________ 211
8 Measurement of Discharge by Miscellaneous Methods ____________ 260
9 Indirect Determination of Peak Discharge _____________----------- 273
VOLUME 2. COMPUTATION OF DISCHARGE
Chapter lo--Discharge Ratings Using Simple Stage-Discharge
Relations
Page
Introduction -------_________________________________---------------------- 285
Stage-discharge controls-------------- _____ -_----------------- ______________ 286
Graphical plotting of rating curves ------- ______________ ------------- ________ 287
Section Controls ---______________---____________________------------------ 294
Artificial controls -- ___________ -------- ______________ ---------_- ________ 294
Transferability of laboratory ratings ___----------- _______________--- 295
Thin-plateweirs -_________---------_____________________~--------- 295
Rectangular thin-plate weir _____ -- _________________ ---------___ 296
Trapezoidal thin-plate weir -------____~---------_______________ 299
Triangular or V-notch thin-plate weir _____-------------________ 303
Submerged thin-plate weirs ------ ____ -------------- ____________ 305
Broad-crested weirs ____________ ----- ________________ --------- ______ 306
Flat-crested rectangular weir ----- __________ ----------- ________ 307
Notched flat-crested rectangular weir _________ ----------- ________ 309
Trenton-type control --------_-___------------- _______________- 311
Columbus-type control _________ ---- ___________________ ---------- 312
Submerged broad-crested weirs ----------------_-______________ 312
Flumes-----------___-~~------------~~~~~~~~~~~~~~~------------~--- 312
Parshallflume -____--------------_____________________-------- 314
Trapezoidal supercritical-flow flume _______._____.___ ~---------- 320
Naturalsectloncontrols___..._~-------______~~~~~~~~~~~~~~.~~---------- 326
Compound section controls_---------- ._______.________________ -~---- 327
III

IV CONTENTS
Chapter lo-Discharge Ratings Using Simple Stage-Discharge Relations-Continued
Page
Channelcontrol-_----____________________---------------------------------- 328
Channel control for stable channels ___________ -_- ______ - __________ --_-- 328
Compound controls involving channel control _______________-________ 330
Extrapolation of rating curves ----- _______ -_--------- ____________ ------- ____ 332
Low-flow extrapolation ________________________________________-------- 333
High-flow extrapolation _________-___________ -_-_-- _______ -_- ____ ------- 334
Conveyance-slope method ______________-_________________________-- 334
Arealcomparisonofpeak-runoffrates -_-__-__-_- _____ --_- _____ ----- 337
Step-backwater method _______ -_-_-_-_---------- _____ -_-__-------_- 338
Flood routing-----____-_--____________________--------------------- 344
Shifts in the discharge rating __-__------ __________ -_-------__-_-_---__----- 344
Detection of shifts in the rating -__-_- ____-__________ -- ____________-____ 345
Rating shifts for artificial controls ----__- ______ -_------___-_-___--_----- 348
Rating shifts for natural section controls _-_-_ -__-_- ____ ---_-__-_- ____ -_- 352
Rating shifts for channel control _---_-- -_-__ - _______ ------ _-____ -_- _____ 354
Effect of ice formation on discharge ratings _________-_--_ -__- _______________- 360
General _______-_-______________________________---------------------- 360
Frazil --~~~~~~----_~~~~~~~____________________~~~-~--~--~~~~~~~~~~~~~~ 360
Anchor ice __-_-__________--_-_____________________-------------------- 361
Surface ice --___-_-~~~~-_-__-_-____________________~--~~~~~~~~~~~~-~--~ 362
Formation of ice cover~~~~~~~~~~~~~~~____________________~~~~~~~~~-~ 362
Effect of surface ice on stream hydraulics------- __-_--_ -_-_- ____ ----- 363
Computation of discharge during periods of backwater from anchor ice ____ 364
Computation of discharge during periods of backwater from surface ice --__ 366
Discharge-ratio method ________-________ -__----- ______ --__--------- 368
Shifting-control method - ________ --_-__----- ______ -_-__------- ______ 369
Hydrographic- and climatic-comparison method _- _____________ -_-___- 370
Sand-channel streams _______-_ -_-_------ _______ -__----- _______ -_---_-----__ 376
Bedconfiguration _-----_________-_ --_- _______-_ ---_-___-- _-_--_ -___-_-_ 377
Relation of mean depth to discharge ___--__________-__ -__- _______-_-_ --_ 379
Development ofdischarge rating_-_-_-_---_--_-__---- ______ --_-__-----__ 382
Evidences of bed forms _-----________-__-_ -_--_- ______-____ -_-___-- 384
Shifting controls ___-__-______-_-_--_------------------------------ 385
Artificial controls for sand channels - _____-____ -__-----_--_-_-- ______ --_ 387
Selectedreferences __-________-_----_-_____________________---------------- 388
Chapter 11-Discharge Ratings Using Slope as a Parameter
Page
General considerations _.___-_--_--_-_-__-_____________________---------~--- 390
Theoretical considerations __-____ -- ____---__________-_-_ - _________ -_- ___-____ 391
Variable slope caused by variable backwater __--__________-__________________ 392
Ratingfallconstant --__-_-____-__-_-__-____________________----------- 396
General discussion of rating principles _____--_ - _______-______________ 396
Procedure for establishing the rating _____--_ - ______-_--_ - ______-___ 398
Example ofrating procedure -__--_--__-_-_____-____________________ 400
Rating fall a function of stage _- ________- --_- __-______ - _________-________ 400
General discussion of rating principles _______ -__- ______ -_--_- ___-____ 400
Procedure for establishing the rating _-_____________________________ 409
Examples of rating procedure __________________ - _______ -_- ______--_- 411
Determination of discharge from relations for variable backwater __--_____ 412

CONTENTS V
Chapter 11-Discharge Ratings Using Slope as a Parameter-Continued
Page
Variable slope caused by changing discharge _________ ------ ____________------- 413
Theoreticalconsiderations______------------------------------------------413
Methods of rating adjustment for changing discharge ________--------- -----416
Bayer method ________________________________________--------------4I6
Wigginsmethod _______________----_____________________------------418
Variable slope caused by a combination of variable backwater and changing dis-charge_____---------------------------------------------------------------
421
Shifts in discharge ratings where slope is a factor---- _________------------- ----422
A suggested new approach for computing discharge records for slqpe stations ----423
Selected references ____________________--~~-~-~------------------------------428
Chapter 12-Discharge Ratings Using a Velocity Index as a
Parameter
Page
Introduction ---------------------------------------------------------------.429
Standard current-meter method ________________________________________-----. 430
Deflection-meter method ----------------------------------------------------432
General -----------________ - ________________________________________---- 432
Vertical-axis deflection vane ________________________________________---- 432
Horizontal-axis deflection vane ________________________________________-- 43~
Examples of stage-velocity-discharge relations based on deflection-meter obser-vations
--------------------------------------------------------------437
Acoustic velocity-meter method --__----- _____________________________________ 439
Description ------------------------------------------------------------439
Theory____-_---_--______________________~~~~~~~~~~~-~~~~~-~~~~~~~~~~-~--441
Effect of tidal flow reversal on relation of mean velocity to line velocity ----448
Orientation effects at acoustic-velocity meter installations~~~~~~~-__________448
Effect of acoustic-path orientation on accuracy of computed line velocity
(V‘) ------_- ____-_-__________--_____________________-------------- 448
Effect of variation in streamline orientation ___________---------------451
Factors affecting acoustic-signal propagation __----------------------------454
Temperature gradients _-_-_-___-_-____________________________------454
Boundary proximity --------------- _______________________________ --454
Air entrainment _-------_---_-___-______________________------------456
Sedimentconcentration______-_-----___________________________~~~~~~456
Aquatic vegetation _-_--_-- _________________________________ ----_----457
Summary of considerations for acoustic-velocity meter installations --------459
Electromagnetic velocity-meter method - _____________________________________ 459
General ---_______-------------------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-~~~~~459
Point-velocity index --______________________________________------------460
Instrumentation _______-_-______________________________-----~-~~~~~460
Analysis of point-velocity data------- _________________________________ 461
Integrated-velocity index --_-----_---________~~~--~-~-~-~-~---~----------4~
Theory ________________________________________---------------------- 464
Instrumentation _------------------_____________________~~~---------465
Appraisal of method --------------- _________________________________ 468
Selectedreferences _-------------------____________________-~~~~~~~~~~~~-----470
Chapter 13-Discharge Ratings for Tidal Streams
Page
General ________________________________________----------------------------471
Evaluation of unsteady-flow equations --- __________ --------- _________--------- 471
Power series ________________________________________--------------------473
Method of characteristics _____________ - ________________------------ ------474

VI CONTENTS
Chapter 13-Discharge Ratings for Tidal Streams-Continued
Page
Evaluation of unsteady-flow equations-Continued
Implicit method _____ ---------------- ____________ -_---------------------475
Fourier series ________________________________________------------------475
Empirical methods _-_-___--------------------------------------~~~--~~~~---.475
Method ofcubatures ________________________________________-----------.476
Rating-fall method ---------------____-~----------:-~--------------------479
Tide-correctionmethod ---------------_________________________---------.479
Coaxial rating-curve method _________________________________ ~~ _______. 481
Selectedreferences ____------------------~~~-~~--~--~-------------------~-~-.4~
Chapter 14-Discharge Ratings for Miscellaneous
Hydraulic Facilities
Page
Inr4oduction.....486
Dams with movable gaates.....486
General.....486
Drum gates.....488
Radial or Tainter gates.....496
Radial gates on a horizontal surface .....497
Radial gates on a curved dam crest or sill ..... 499
Vertical lift gates ..... 507
Roller gates ..... 508
Movable dams ..... 508
Flash boards ..... 512
STop logs and needles ..... 514
Navigation locks ..... 514
Measurement of leakage through navigation locks ..... 515
Pressure conduits ..... 520
General ..... 520
Metering devices for pressure-conduit flow ..... 521
Mechanical meters ..... 521
Differential-head meters /..... 522
Electormagnetic velocity meter ..... 528
Acoustic fvelocity meter ..... 528
Laser flowmeter ..... 529
Discharge-measurement nmethods for meter calibration ..... 529
Measurement of discharge by pitot-static tubes and pitometers ..... 529
Measurment of discharge by salt-velocity method ..... 533
Measurement of dischage by the Gibson method .... 533
Calibration of turbines, pumps, gates, and valves .... 536
Urban storm drains ..... 538
Selected references ..... 542
Chapter 15-Computation of Discharge Records
Page
General -----------------_______________________---~-~-~~~-~~~~~~~~~~~~~~~ 544
Station analysis----------______________________---------------------------- 544
Datumcorrections -------------~-~--_-~-_----------------------------- 545
Review of discharge measurements_____--------------------------------- 547
Station rating-simple stage-discharge relation _______~_~--_~--_---_--___ 549
Plotting of discharge measurements _------------------------------- 549

CONTENTS
Chapter 15-Computation of Discharge Records-Continued
VII
Page
Station analysis-Continued 550
Station rating-simple stage-discharge relation-Continued 555
Station rating-three-parameter discharge relation __---___---____-______ 558
Computation of discharge records for a nonrecording gaging station _________- 559
Computation of gage-height record -__-- _________ --___----__-----_--- ____ 559
Computation of discharge records for a recording station equipped with a graphic
recorder _----_----________--____________________--~~~----~------~--~---- 560
Computation of gage-height record __---__--__----__--- ____ -- ___________- 560
Determinationoftimecorrections _-- ________ - _____ --_----___-- _____ 560
Determination of gage-height corrections ___---__----__-- ___________- 563
Determination of daily mean gage height __----_-----__---___________ 564
Subdivision ofdaily gage heights_------ ____ ---------------------~--- 564
Computation of daily discharge --_----_--___--------------- ___________- 569
Preparation of form for computing and tabulating discharge ___-______ 569
Determination of discharge from the gage-height record _----__---____ 571
Estimation of daily discharge for periods of indeterminate stage-discharge
relation---__------------~---__-----~~~----~~----~~--~---~ 572
Estimation of daily discharge for periods of no gage-height record ____ 573
Case A. No gage-height record during a low- or medium-flow reces-sion
on an uncontrolled stream---- _____ - _____ --___----_-----__ 574
Case B. No gage-height record during periods of fluctuating dis-charge
on an uncontrolled stream _-----_---___--- _____ - ______ 575
Case C. No gage-height record for a station on a hydroelectric pow-erplantcanal_______---___-----~-~----~~~-----~-----~~------~
577
Case D. No gage-height record for a station immediately downsteam
froma reservoir.__-----_------__-----------------------~------- 578
Case E. No gage-height record for a station on a controlled stream
where the station is far downstream from the known controlled
release__--________----__------~-~~~--~~~----------~----~---~ 578
Completionofthedischargeform ___-- ________ --___--___- ___________--_ 579
Record of progress of discharge computations _----_----___-- ___________-_ 580
Station-analysis document-_----___--------------~--- ________________--_ 580
Station an&q& ______________________- __________________ ___________--------- -------- ------ +Cjg2
Computation of discharge records when a three-parameter discharge relation
isused--~-~--_-~-_-----_------------~-~~~~---~~~~~--~~-~---~-------- 586
Computation of discharge records for a recording station equipped with a
digital recorder ---_-______-_____---____________________--~~~~-~-~~~~-~-~ 587
General __---~______----__---____________________~~~--~~-~----------~ 587
Input to computer_____---___-----__---_--___----__~----~------~-----~- 588
Output from computer_---___-----______---~--~~~~-~-~~~~---~~----------~-- 588
Sequence of operation of an automated computing system __---___---_ ___ 592
Selected references _--__----__---____--____________________~-----~~---~---- 599
Chapter 16-Presentation and Publication of Stream-Gaging Data
General.........................................................................................................................601
Format............................................................................................................................601
Selected reference...........................................................................................................603
Index

VIII CONTENTS
ILLUSTRATIONS
F
FIGURE 139. Example of form used for tabulating and summarizing
current-meter discharge measurements ____________-_
140. Example showing how the logarithmic scale of graph paper
maybe transposed _--_____-_________________________
141. Rating-curve shapes resulting from the use of differing val-uesofeffectivezeroflow
____ -----_----__----__--____
142. Schematic representation of the linearization of a curve on
logarithmicgraphpaper __-__--- _____ -___---___--___
143. Definition sketch of a rectangular thin-plate weir _-________
144. Discharge coefficients for full-width, vertical and inclined,
rectangular thin-plate weirs __________-_____--______
145. Definition of adjustment factor, k., for contracted rectangular
thin-plateweirs ___---___-__--___----~~----------~-~
146. Rating curve for hypothetical rectangular thin-plate weir --
147. Sketch of upstream face of a trapezoidal weir ______-_____--
148. Sketch of upstream face of a triangular or V-notch weir _---
149. Sketch showing submergence of a weir - ____ - ______________
150. Generalized relation of discharge ratio to submergence ratio
forverticalthin-plate weirs ___----__---____--________
151. Coefficients of discharge for full-width, broad-crested weirs
with downstream slops s.1: 1 and various upstream slopes
152. Sketch of upstream face of flat-crested weir with sloping crest
andcatenarycrest ___- ____ --_----__--- _____ - ______-
153. Rating curve for a notched broad-crested control at Great
Trough Creek near Marklesburg, Pa - - - - __ ~ - - _ _ 154. Cross section of Trenton-type control--- ____________--___--
155. Dimensions of Columbus-type control - ____ - ____________--
156. Configuration and descriptive nomenclature for Parshall
flumes ---___---___--____-_----------~-----~----~--
157. Discharge ratings for “inch” Parshall flumes for both free-flow
and submergence conditions _-----_-----___--___
158. Correction factors for submerged flow through l- to 50-ft Par-shall
flumes __________-___--__--___________________
159. Configuration and dimensions of trapezoidal supercritical-flow
flumes ofthree throat widths __---~_-----_----___
160. Sketch illustrating use of the total-energy (Bernoulli) equa-tion__-----_----------~--------~-----~~----~~-----~-
161. Stage-discharge relation and significant depth-discharge re-lations
for l-ft trapezoidal supercritical-flow flume _ _ 162. Stage-discharge relation and significant depth-discharge re-lations
for 3-ft trapezoidal supercritical-flow flume ____
163. Stage-discharge relation and significant depth-discharge re-lations
for 8-ft trapezoidal supercritical-flow flume --__
164. Rating curve for a compound section control at Muncy Creek
nearSonestown,Pa _- ____ -__----__----__---________
165. Rating curve for a compound control at Susquehanna River
atHarrisburg,Pa ___-_______-----------~------~----
166. Example of low-flow extrapolation on rectangular-coordinate
graph paper ___________-__--__--____________________
167. High-flow extrapolation by use of conveyance-slope
method-Klamath River at Somes Bar, Calif -__-----__
‘age
288
290
292
293
297
298
298
301
302
304
305
306
308
309
310
311
313
314
319
320
321
323
324
325
325
328
331
333
336

CONTENTS IX
FIGURE 168. Relation of peak discharge to drainage area and maximum
24-hour basinwide precipitation in north coastal
California, December 1964 __----___--- ______________
169. Dimensionless relation for determining distance required for
backwaterprofilestoconverge _----__-----_----_____
170. Rating curve for hypothetical rectangular thin-plate weir,
with shift curves for scour and fill in the weir pool ____
171. First example of a stage-shift relation and the corresponding
stage-discharge relation caused by scour or till in the
controlchannel --__----__- ____ ----__----__---- _____
357
362
364
172. Second example of a stage-shift relation and the correspond-ing
stage-discharge relation caused by scour or till in the
controlchannel -___---___-__-----__----------- ____ -
173. Typical anchor-ice rises ___- _____ - ___________________-----
174. Typical rise as complete ice cover forms _______-_____---__
175. Effect of siphon action at artificial control in Sugar Run at
Pymatuning, Pa., January 4-5,1940_- ___________---__
176. Rating curve for Menominee River near Pembine, Wis ____
177. Example of discharge-ratio method for correcting discharge
record for ice effect_-_____---------------------------
178. Example of shifting-control method for adjusting stage record
foriceeffect ----___---____-__---____________________
179. Daily hydrographs for open-water discharge and for dis-
365
366
367
370
charge corrected for ice effect ___---___-----_-----____ 375
180. Comparison of daily winter discharge at two gaging stations
showing their response to air-temperature fluctuations
181. Idealized diagram of bed and water-surface configuration of
alluvial streams for various regimes of flow ______--__
182. Typical loop curve of stage versus discharge for a single flood
event in a sand channel ___-_ --- _____ - ____________--__
183. Stage-discharge relation for Huerfano River near Under-cliffe,
Colo---- _____________ ---- ____ - ___________---__
184. Relation of velocity to hydraulic radius for Huerfano River
nearUndercliffe,Colo ______ -----__----___--- _______
185. Relation of velocity to hydraulic radius for Rio Grande near
Bernallilo,N.Mex _- _______ --- ____ - _________________
186. Stage-discharge relation for station 34 on Pigeon Roost
Creek,Miss _-----_-----_______---~~------~----~-~--
187. Relation of stream power and median grain size to form of
bedroughness _____-_______-----____________________
188. Schematic representation of typical stage-fall relations ____
189. Schematic representation of family of stage-discharge curves,
each for a constant but different value of fall -_________
190-195. Stage-fall-discharge relations for:
190. Tennessee River at Guntersville, Ala __-- ________
191. Columbia River at the Dalles, Oreg ___-____---_--
192. Ohio River at Metropolis, Ill --__--- _________-___
193. Kelly Bayou near Hosston, La ---- ____ -~ ____-___
194. Colusa Weir near Colusa, Calif ---- _________--___
195. Kootenay River at Grohman,
British Columbia, Canada -- __________---_--
339
341
350
356
376
378
380
380
381
382
383
386
394
397
401
402
403
404
406
407

X CONTENTS
FIGURE 196.
197.
198.
199.
200.
201.
202.
203.
204.
205.
206.
207.
208.
209.
210.
211.
212.
213.
214.
215.
216.
217.
218.
219.
220.
221.
222.
223.
224.
Page
Stage-discharge loop for the Ohio River at Wheeling, W. Va.,
during the flood of March 14-27,1905 _.___._. ___--._. 413
Adjustment of discharge measurements for changing dis-charge,
Ohio River at Wheeling, W. Va, during the
period March 14-27, 1905 __--- ____ ~_-- ________---__ 417
Diagrams for solution of the Manning equation to determine
S, when A n= 0.025, B n=0.035, C n= 0.050, and D
n=0.080________----___----------------------------- 420
Diagram for determining slope increment resulting from
changing discharge--_-- _____ --- ____ ---- ______ ______ 424
Diagrams for determining factor to apply to measured dis-charge
for rising stage and falling stage.-------------- 426
Hypothetical relation of mean velocity in measurement cross
section to stage and index velocity _____________--- ~_ 431
Sketch of two types of vertical-axis deflection vanes -_ __ _ 433
Plan and front-elevation views of a vertical-axis deflection
meter attached to a graphic recorder ___________----_ 434
Sketch of a pendulum-type deflection vane ___________---- 436
Calibration curve for pendulum-type deflection vane _----~ 437
Recorder chart for a deflection-meter gaging station on a
tidal stream ____--______----______________________ 438
Rating curves for a deflection-meter gaging station on a tidal
stream _-------_~_-----_________________________- 439
Rating curves for a deflection-meter gaging station on Lake
Winnipesaukee outlet at Lakeport, N.H - _________ ~___ 440
Transducer _-------_~~-----__---~____________________ 442
Console ------------____~---_____________________~~--.-- 443
Sketch to illustrate operating principles of the acoustic veloc-itymeter
------_~------___-_______________________~ 444
Relation between stage and mean-velocity coefficient, K, for
the acoustic-velocity meter (AVMJ system, Columbia
River at The Dalles, Ore ____ ----__~~--- __.___ -- _____ 447
Relation between C!, and velocity and tide phase _~---_____ 450
Possible variation in streamline orientation -__~-----_____ 453
Curves used as a preliminary guide for AVM site selection,
based solely on consideration of channel geometry _.__ 455
Interrelation between signal strength, sediment concentra-tion,
particle size, and acoustic-path length -----_____ 458
Electromagnetic probe, model 201, Marsh-McBnney _--~~_ 462
Relation between point-index velocity and mean stream vel-ocity
for Alabama River near Montgomery, Ala -_____ 463
Instrumentation for an electromagnetic stream-gaging sta-tion----_____._-________-~---------~---------~---.--
466
Schematic diagram showmg inclusion of bed and bank mate-rial
in the stream cross section __----~_._---------~-- 467
Block diagram showing the function of the data processor-. 469
Sample computation of tide-affected discharge by method of
cubatures, using 30-minute time intervals _._~~-~_____ 477
Discharge hydrograph obtained for sample problem by
methodofcubatures ----____-~__--------~-________~ 478
Graph of relation between tide-corrected gage height and dis-charge
for Miami Canal at Water Plant, Hialeah, Fla. _ 480

CONTENTS XI
FIGURE 225. Stage and discharge of the Sacramento River at Sacramento,
Calif., Sept. 30 to Oct. 1, 1959 __~----___-------__--- 482
226. Coaxial rating curves for the Sacramento River at Sac-ramento,
Calif___-----------___~-------------------- 483
227. Two types of drum gates _~---___---------_______________ 488
228. Drum-gate positions __-----__----- _____ --_------ _____ -__ 489
229. General curves for the determination of discharge coefficients 490
230. Plan of Black Canyon Dam in Idaho--------__------------ 491
231. Spillway crest detail, Black Canyon Dam, Idaho -----__--- 492
232. Diagram for determining coefficients of discharge for heads
233.
234.
235.
236.
237.
238.
239.
240.
241.
242.
243.
244.
245.
other than the design head -- _____ ----_ __------___--- 493
Head-coefficient curve, Black Canyon Dam, Idaho ________ 495
Relation of gate elevation to angle 8 ----_-----__________ 496
Rating curves for drum-gate spillway of Black Canyon Dam,
Idaho __-----_________---_____________________--~--- 498
Cross-plotting of values from initial rating curves, Black
CanyonDam,Idaho -----__-______---_______________ 499
Definition sketch of a radial gate on a horizontal surface -- 500
Coefficient of discharge for free and submerged efllux, a/r =
0.1 ----_----____--------~--~~~~~~---~~~~~-------~-- 501
Coefficient of discharge for free and submerged efflux, u/r =
0.5 _-----________------____________________~~--~-~~ 502
Coefficient of discharge for free and submerged efflux, a/r =
0.9 ---------- _____ ~-----__~ _______ ~- ______ -----_--- 503
Definition sketch of a radial or Tamter gate on a sill ______ 504
Schematic sketches of roller gates__---------------------- 508
Bear-trap gate-~-______~-----___---------------~-------~ 509
Hinged-leaf gate - ______ ------ ________ ----___-------__--- 510
Wickets __-----__________~--_____________________------- 511
246. Discharge coefficients for an inclined rectangular thin-plate
welr--_--------------------------------------------- 512
247. Flashboards ___________~_____________________________- 513
248. Definition sketch of a lock _____ ------ ____ ----- __________ 516
249. Storage diagram starting with lock chamber full __________ 517
250. Three-types of constrxtion meter for pipe flow -----~~~_--- 523
251. Discharge coefficients for venturi meters as related to
Reynolds number -----__-----_______~______________ 525
252. Schematic view of one type of electromagnetic velocity meter 528
253. Schematic drawing of pitot-static tube and Cole pitometer 530
254. Locations for nitot-tube measurements in circular and
255. Sample record of a salt cloud passing upstream and
downstream electrodes in the salt-velocity method of
measuring flows in pipelines ___-----_-______----___ 533
256. General arrangement of salt-velocity equipment for pressure
conduits-~-----__~-------~-------~-----------~-----~ 534
257. Brine-Injection equipment rn conduit ----___-____----_--- 535
258. Gibson apparatus and pressure-variation chart-------~---~ 536
259. Sketch of USGS flowmeter in a sewer --___---~~-----~---- 539
260. Sketch of Wenzel asymmetrical flowmeter in a sewer __---_ 542

XII CONTENTS
Page
FIGURE 261. Level notes for check of gage datum ____-_________________ 546
262. List of discharge measurements -__-_-__- ______ - ____ - _____ 548
263. Logarithmic plot of rating curve -----__---_------__----- 551
264. Rectangular plot of low-water rating curve _-____-___---___ 552
265. Standard rating table _-_-___---___-___- _____ - __________ 556
266. Expanded rating table _---___--- ____ _ __________________ 557
267. Computation of daily mean gage height on graphic-recorder
chart __--___--___---__---____________________------ 561
268. Example of graphical interpolation to determine time correc-tions
-__-_--_---___--___-____________________------ 562
269. Definition sketch illustrating computation of stage limits for
application of discharge _____________--___---------- 565
270. Results of computation of allowable limits of stage for Rating
no. 4, Clear Creek near Utopia, Calif ___-___________- 566
271. Table of allowable rise for use with Rating no. 4, Clear Creek
near Utopia, Calif __________________________________ 566
272. Sample computation of daily mean discharge for a subdivided
day by point-intercept method ________--__----__---- 568
273. Computation of daily discharge _______-____--__---_______ 570
274. Form showing progress of computation of graphic-recorder
record__---__----------__-----------~--~-------------- 581
275. Correction and update form for daily values of discharge __ 589
276. Primary computation sheet for routine gaging station ---- 590
277. Primary computation sheet for slope station ___-______--__ 591
278. Primary computation sheet for deflection-meter station ---_ 593
279. Printout of daily discharge - ____ - _______________-____---- 594
280. Digital-recorder inspection form--. _________-____---__---- 595
281. Printout from subprogram for updating primary computation
sheet ___-___-----_---___-____________________------ 598
282. Form showing progress of computation of digital-recorder
record(sample l)__----__----__---_-_______-_____---_ 599
283. Form showing progress of computation of digital-recorder
record(sample 2)__---__-----_----_______________---_ 600
284. Table of contents for annual published report ___--___---- 604
285. List of surface-water stations ---__----__-- ____ - ______-___ 605
286. Introductorytextpages_______________-____-___----__-------- 606
287. Map of gaging-station locations --_----__-- ___________---- 615
288. Bar graph of hydrologic conditions _____________---__---- 616
289. Daily discharge record __---___-- ____ - ____ - ______________ 617
290. Daily discharge record (adjusted) __--- ____ - _____-____---- 618
291. Daily reservoir record _----__---___-- ____ - _____--___---- 619
292. Monthly reservoir record ---____-__________-_____________ 620
293. Group reservoir records (large reservoirs) __---__--------- 621
294. Group reservoir records (small reservoirs) _______-____---- 623
295. Discharge tables for short periods __________--___---__---- 624
296. Revisions of published records --___--__---___--___-- ____ 625
297. Schematic diagram showing reservoirs, canals, and gaging
stations _----_----___--____-____________________---- 626
298. Low-flow partial records _-- ____ - _________ - ______________ 627
299. Crest-stage partial records - ____ -___-- ____ - ___________--_ 628
300. Discharge measurements at miscellaneous sites _______--- 629

CONTENTS XIII
Page
FIGURE 301. Seepage investigation __-- ________ - _____ ----_----_-_--__ 629
.302. Low-flow investigation ________ ---___---_- _________--___- 630
303. Index for annual published report __- ____ ----__--- __-_____ 631
TABLES
TABLE 16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
Page
Computation of discharge rating for a hypothetical
rectangular thin-plate weir __-___--- _________________ 300
Dimensions and capacities of all sizes of standard Parshall
flumes -__------_--_____--_____________________------ 315
Discharge table for Parshall flumes, sizes 2 inches to 9 inches,
for free-flow conditions __--__--- ____ ---__---- ____ -___ 317
Discharge table for Parshall flumes, sizes 1 foot to 50 feet, for
free-flow conditions __________ ------- _____ - ___________ 318
Hypothetical stage-discharge rating table for a compound con-trol
_-____----__----___-____________________---~---- 332
Surface and bed descriptions for the various flow regimes -- 379
Variation of Cp with tidal phase --_-_---- _________________ 449
Error in computed V,, attributable to resolution error, for various
acoustic-pathorientations, foragiven AVM system -_____ 451
Ratio of computed discharge to true discharge for various
combinations of 8 and ~------------------------------ 454
Head and discharge computations for a free crest (Black Can-yon
Dam in Idaho) --___---__---___---_____ __________ 494
Head and discharge computations for drum gates in raised
positions ----_-_-----___----_____________________---- 497
Values of kinematic viscosity corresponding to selected water
temperatures --___----__---__________________________ 524
CONVERSION FACTORS
[Factorsf or convertingI nch-poundto metric units me shownt o four sigmlieantf igures However.m the text the
metric equivalents,w heres hown.a re camedo nly to the numbero f swnnkant figuresc onsistenwt th the values
for the English umts ]
Inch-pound Multrply by- M‘AW
acres 4.047 x lo:’ m’ (square meters)
acre-ft (acre-feet) 1.233x lo” m:’ (cubic meters)
acre-ft/yr (acre feet per year) 1.233x lo” m”/yr (cubic meters per year)
ft (feet) 3.048x 10-l m (meters)
ft/hr (feet per hour) 3.048x 10-l m/hr (meters per hour)
ft/s (feet per second) 3.048x 10-l m/s (meters per second)
ft:Ys (cubic feet per second) 2.832 x lo-’ m”/s (cubic meters per second)
in (inches) 2.540x 10 mm (millimeters)
lb (pounds) 4.536x lo-’ kg (kilograms)
mi (miles) 1.609 km (kilometers)
mi” (square miles) 2.590 km2 (square kilometers)
02 (ounces) 2.835x lo-’ kg (kilograms)

MEASUREMENT AND COMPUTATION
OF STREAMFLOW
VOLUME 2. COMPUTATION OF DISCHARGE
By S. E. KANTZ and others
CHAPTER IO.-DISCHARGE RATINGS USING SIMPLE
STAGE-DISCHARGE RELATIONS
INTRODUCTION
Continuous records of discharge at gaging stations are computed by
applying the discharge rating for the stream to records of stage. Dis-charge
ratings may be simple or complex, depending on the number of
variables needed to define the stage-discharge relation. This chapter
is concerned with ratings in which the discharge can be related to
stage alone. (The terms “rating,” “rating curve,” “stage rating,” and
“stage-discharge relation” are synonymous and are used here inter-changeably.)
Discharge ratings for gaging stations are usually determined em-pirically
by means of periodic measurements of discharge and stage.
The discharge measurements are usually made by current meter.
Measured discharge is then plotted against concurrent stage on graph
paper to define the rating curve. At a new station many discharge
measurements are needed to define the stage-discharge relation
throughout the entire range of stage. Periodic measurements are
needed thereafter to either confirm the permanence of the rating or to
follow changes (shifts) in the rating. A minimum of 10 discharge
measurements per year is recommended, unless it has been demon-strated
that the stage-discharge relation is unvarying with time. In
that event the frequency of measurements may be reduced. It is of
prime importance that the stage-discharge relation be defined for
flood conditions and for periods when the rating is subject to shifts as
a result of ice formation (see section titled, “Effect of Ice Formation on
Discharge Ratings”) or as a result of the variable channel and control
conditions discussed in the section titled, “Shifts in the Discharge
Rating.” It is essential that the stream-gaging program have suffi-cient
flexibility to provide for the nonroutine scheduling of additional
measurements of discharge at those times.
If the discharge measurements cover the entire range of stage ex-perienced
during a period of time when the stage-discharge relation is
stable, there is little problem in defining the discharge rating for that
285

286 COMPUTATION OF DISCHARGE
period. On the other hand, if, as is usually the case, discharge meas-urements
are lacking to define the upper end of the rating, the defined
lower part of the rating curve must be extrapolated to the highest
stage experienced. Such extrapolations are always subject to error,
but the error may be reduced if the analyst has a knowledge of the
principles that govern the shape of rating curves. Much of the mate-rial
in this chapter is directed toward a discussion of those principles,
so that when the hydrographer is faced with the problem of extending
the high-water end of a rating curve he can decide whether the ex-trapolation
should be a straight line, or whether it should be concave
upward or concave downward.
The problem of extrapolation can be circumvented, of course, if the
unmeasured peak discharge is determined by use of the indirect
methods discussed in chapter 9. In the absence of such peak-discharge
determinations, some of the uncertainty in extrapolating the rating
may be reduced by the use of one or more of several methods of
estimating the discharge corresponding to high values of stage. Four
such methods are discussed in the section titled “High-flow Extrapo-lation.”
In the discussions that follow it was generally impractical to use
both English and metric units, except where basic equations are giv-en.
Consequently English units are used throughout, unless other-wise
noted.
STAGE -DISCHARGE CONTROLS
The subject of stage-discharge controls was discussed in detail in
chapter 3, but a brief summary at this point is appropriate.
The relation of stage to discharge is usually controlled by a section
or reach of channel downstream from the gage that is known as the
station control. A section control may be natural or manmade; it may
be a ledge of rock across the channel, a boulder-covered riffle, an
overflow dam, or any other physical feature capable of maintaining a
fairly stable relation between stage and discharge. Section controls
are often effective only at low discharges and are completely sub-merged
by channel control at medium and high discharges. Channel
control consists of all the physical features of the channel that deter-mine
the stage of the river at a given point for a given rate of flow.
These features include the size, slope, roughness, alinement, constric-tions
and expansions, and shape of the channel. The reach of channel
that acts as the control may lengthen as the discharge increases,
introducing new features that affect the stage-discharge relation.
Knowledge of the channel features that control the stage-discharge
relation is important. The development of stage-discharge curves
where more than one control is effective, and where the number of

SIMPLE STAGE-DISCHARGE RELATIONS 287
measurements is limited, usually requires judgment in interpolating
between measurements and in extrapolating beyond the highest
measurements. That is particularly true where the controls are not
permanent and the various discharge measurements are representa-tive
of changes in the positioning of segments of the stage-discharge
curve.
GRAPHICAL PLOTTING OF RATING CURVES
Stage-discharge relations are usually developed from a graphical
analysis of the discharge measurements plotted on either
rectangular-coordinate or logarithmic plotting paper. In a prelimi-nary
step the discharge measurements available for analysis are
tabulated and summarized on a form such as that shown in figure
139. Discharge is then plotted as the abscissa, corresponding gage
height is plotted as the ordinate, and a curve or line is fitted by eye to
the plotted points. The plotted points carry the identifying measure-ment
numbers given in figure 139; the discharge measurements are
numbered consecutively in chronological order so that time trends
can be identified.
At recording-gage stations that use stilling wells, systematic and
significantly large differences between inside (recorded) gage heights
and outside gage heights often occur during periods of high stage,
usually as a result of intake drawdown (see section in chapter 4 titled,
“Stilling Wells”). For stations where such differences occur, both in-side
and outside gage heights for high-water discharge meas-urements
are recorded on the form shown in figure 139, and in plot-ting
the measurements for rating analysis, the outside gage readings
are used first. The stage-discharge relation is drawn through the
outside gage readings of the high-water discharge measurements and
is extended to the stage of the outside high-water marks that are
observed for each flood event. The stage-discharge relation is next
transposed to correspond with the inside gage heights obtained from
the stage-recorder at the times of discharge measurement and at flood
peaks. It is this transposed stage-discharge relation that is used with
recorded stages to compute the discharge.
The rationale behind the above procedure is as follows. The outside
gage readings are used for developing the rating because the hydrau-lic
principles on which the rating is based require the use of the true
stage of the stream. The transposition of the rating to inside (re-corded)
stages is then made because the recorded stages will be used
with the rating to determine discharge. The recorded stages are used
for discharge determination because if differences exist between in-side
and outside gage readings, those differences will be known only
for those times when the two gages are read concurrently. If the

: : : : :
:

outside gage heights were used with the rating to determine dis-charge,
variable corrections, either known or assumed, would have to
be applied to recorded gage heights to convert them to outside stages.
We have digressed here to discuss differences between inside and
outside gage heights, because in the discussions that follow no dis-tinction
between the two gages will be made.
The use of logarithmic plotting paper is usually preferred for
graphical analysis of the rating because in the usual situation of
compound controls, changes in the slope of the logarithmically plotted
rating identify the range in stage for which the individual controls
are effective. Furthermore, the portion of the rating curve that is
applicable to any particular control may be linearized for rational
extrapolation or interpolation. A discussion of the characteristics of
logarithmic plotting follows.
The measured distance between any two ordinates or abscissas on
logarithmic graph paper, whose values are printed or indicated on the
sheet by the manufacturer of the paper, represents the difference
between the logarithms of those values. Consequently, the measured
distance is related to the ratio of the two values. Therefore, the dis-tance
between pairs of numbers such as 1 and 2, 2 and 4, 3 and 6, 5
and 10, are all equal because the ratios of the various pairs are identi-cal.
Thus the logarithmic scale of either the ordinates or the abscissas
is maintained if all printed numbers on the scale are multiplied or
divided by a constant. This property of the paper has practical value.
For example, assume that the logarithmic plotting paper available
has two cycles (fig. 1401, and that ordinates ranging from 0.3 to 15.0
are to be plotted. If the printed scale of ordinates is used and the
bottom line is called 0.1, the top line of the paper becomes 10.0, and
values between 10.0 and 15.0 cannot be accommodated. However, the
logarithmic scale will not be distorted if all values are multiplied by a
constant. For this particular problem, 2 is the constant used in figure
140, and now the desired range of 0.3 to 15.0 can be accommodated.
Examination of figure 140 shows that the change in scale has not
changed the distance between any given pair of ordinates; the posi-tion
of the ordinate scale has merely been transposed.
We turn now to a theoretical discussion of rating curves plotted on
logarithmic graph paper. A rating curve, or a segment of a rating
curve, that plots as a straight line of logarithmic paper has the equa-tion,
Q =p(G - eP, (53)
where
Q is discharge;
(G - e) is head or depth of water on the control-this value is
indicated by the ordinate scale printed by the manufacturer or

290 COMPUTATION- OF DISCHARGE
2 X original
FIGURE 140.-Example showing how the logarithmic scale of graph paper may be
transposed.

by the ordinate scale that has been transposed, as explained in
the preceding paragraph;
G is gage height of the water surface;
e is gage height of zero flow for a section control of regular shape,
or the gage height of effective zero flow for a channel control or
a section control of irregular shape;
p is a constant that is numerically equal to the discharge when
the head (G - e) equals 1.0 ft or 1.0 m, depending on whether
English or metric units are used; and
N is slope of the rating curve. (Slope in equation 53 is the ratio of
the horizontal distance to the vertical distance. This uncon-ventional
way of measuring slope is necessary because the
dependent variable Q is always plotted as the abscissa.)
We assume now that a segment of an established logarithmic rat-ing
is linear; and we examine the effect on the rating of changes to the
control. If the width of the control increases, p increases and the new
rating will be parallel to and to the right of the original rating. If the
width of the control decreases, the opposite effect occurs; p decreases
and the new rating will be parallel to and to the left of the original
rating. If the control scours, e decreases and the depth (G - e) for a
given gage height increases; the new rating moves to the right and
will no longer be a straight line but will be a curve that is concave
downward. If the control becomes built up by deposition, e increases
and the depth (G - e) for a given gage height decreases; the new
rating moves to the left and is no longer linear but is a curve that is
concave upward.
When discharge measurements are originally plotted on
logarithmic paper, no consideration is given to values of e. The gage
height of each measurement is plotted using the ordinate scale pro-vided
by the manufacturer or, if necessary, an ordinate scale that has
been transposed as illustrated in figure 140. We refer now to figure
141. The inside scale (e = 0) is the scale printed by the paper manu-facturer.
Assume that the discharge measurements have been plotted
to that scale and that they define the curvilinear relation between
gage height (G) and discharge (Q) that is shown in the topmost curve.
For the purpose of extrapolating the relation, a value of e is sought,
which when applied to G, will result in a linear relation between (G -
e) and Q. If we are dealing with a section control of regular shape, the
value of e will be known; it will be the gage height of the lowest point
of the control (point of zero flow). If we are dealing with a channel
control or section control of irregular shape, the value of e is the gage
height of effective zero flow. The gage height of effective zero flow is
not the gage height of some identifiable feature on the irregular sec-tion
control or in the channel but is actually a mathematical constant

that is considered as a gage height to preserve the concept of a
logarithmically linear head-discharge relation. Effective zero flow is
usually determined by a method of successive approximations.
In successive trials, the ordinate scale in figure 141 is varied for e
values of 1,2 and 3 ft, each of which results in a different curve, but
each new curve still represents the same rating as the top curve. For
example, a discharge of 30 ft”/s corresponds to a gage height (G) of 5.5
ft on all four curves. The true value of e is 2 ft, and thus the rating
plots as a straight line if the ordinate scale numbers are increased by
that value. In other words, while even on the new scale a discharge of
30 ft”/s corresponds to a gage height (G) of 5.5 ft, the head or depth on
the control for a discharge of 30 ft31s is (G-e), or 3.5 ft; the linear
rating marked e = 2 crosses the ordinate for 30 ft3/s. at 5.5 ft on the
new scale and at 3.5 ft on the manufacturer’s, or inside, scale. If
values of e smaller than the true value of 2 ft are used, the rating
curve will be concave upward, if values of e greater than 2 ft are used,
the curve will be concave downward. The value of e to be used for a
rating curve, or for a segment of a rating curve, can thus be deter-mined
by adding or subtracting trial values of e to the numbered
scales on the logarithmic plotting paper until a value is found that
results in a straight-line plot of the rating. It is important to note that
if the logarithmic ordinate scale must be transposed by multiplication
or division to accommodate the range of stage to be plotted, that
transposition must be made before the ordinate scale is manipulated
for values of e.
13 12 11 10
k
; 10987
-
I5; 8 7 6 5
G 7654
I
E 654
d
5 4
4
DISCHARGE,I N CUBIC FEET PER SECON- D
FIGURE 141.-Rating-curve shapes resulting from the use of differing values of effec-tive
zero flow.

A more direct solution for e, as described by Johnson (1952) is
illustrated in figure 142. A plot of G versus Q has resulted in the
solid-line curve which is to be linearized by subtracting a value of e
from each value of G. The part of the rating between points 1 and 2 is
chosen, and values of G,, Gr, Q , and Q? are picked from the coordinate
scales. A value of QzI is next computed, such that
From the solid-line curve, the value of G:, that corresponds to Qn is
picked. In accordance with the properties of a straight line on
logarithmic plotting paper,
(G:, - eY = (G, - e) (G, - e). (54)
Expansion of terms in equation 54 leads to equation 55 which pro-vides
a direct solution for e.
e=
G,G, - G:,’
G, + Gz - 2G:,
(55)
A logarithmic rating curve is seldom a straight line or a gentle
curve for the entire range in stage. Even where a single cross section
of the channel is the control for all stages, a sharp break in the
G1
G3
G2
0 2
/
r
e /
/
I
/
0
Q2 Ql
’ Q; = Q, Q,
(G3 - d2 = (G1 - e) (G2 - e)
G1 G2 - G”3
e = G1 + G2 - 2G3
FIGURE 142.-Schematic representation of the linearization of a curve on
logarithmic graph paper.

294 COMPUTATICn OF DISCHARGE
contour of the cross section, such as an overflow plain, will cause a
break in the slope of the rating curve. Commonly, however, a break in
slope is due to the low-water control being drowned out by a
downstream section control becoming effective or by channel control
becoming effective.
The use of rectangular-coordinate paper for rating analysis has
certain advantages, particularly in the study of the pattern of shifts
in the lower part of the rating. A change in the low-flow rating at
many sites results from a change in the elevation of effective zero flow
(e), which means a constant shift in gage height. A shift of that kind is
more easily visualized on rectangular-coordinate paper because on
that paper the shift curve is parallel to the original rating curve, the
two curves being separated by a vertical distance equal to the change
in the value of e. On logarithmic paper the two curves will be sepa-rated
by a variable distance which decreases as stage increases. A
further advantage of rectangular-coordinate paper is the fact that the
point of zero flow can be plotted directly on rectangular-coordinate
paper, thereby facilitating extrapolation of the low-water end of the
rating curve. That cannot be done on logarithmic paper because zero
values cannot be shown on that type of paper.
As a general rule logarithmic plotting should be used initially in
developing the general shape of the rating. The final curve may be
displayed on either type of graph paper and used as a base curve for
the analysis of shifts. A combination of the two types of graph paper is
frequently used with the lower part of the rating plotted on an inset of
rectangular-coordinate paper or on a separate sheet of rectangular-coordinate
paper.
SECTION CONTROLS
ARTIFICIAL CONTROLS
At this point we digress from the subject of logarithmic rating
curves to discuss the ratings for artificial section controls. A knowl-edge
of the rating characteristics of controls of standard shape is
necessary for an understanding of the rating characteristics of natu-ral
controls, almost all of which have irregular shapes. On pages that
follow we first discuss thin-plate weirs, then broad-crested weirs, and
finally flumes.
Thin-plate weirs are generally used in small clear-flowing streams,
particularly where high accuracy is desired and adequate mainte-nance
can be provided, as in small research watersheds. Flumes are
preferred for use in small streams and canals that carry sediment and
debris, and in other situations where the head loss (backwater) asso-ciated
with a thin-plate weir is unacceptable. Most types of flume may
also be used under conditions of submergence, as opposed to free-flow

conditions, thereby permitting them to operate with even smaller
head loss but with some loss of accuracy of the stage-discharge rela-tion.
The broad-crested weirs are commonly used in the larger
streams.
~I‘KANSFt:KABlI~I’1‘Y OF LABORATORY KATINGS
Standard shapes or dimensions are commonly used in building ar-tificial
controls, and many of these standard structures have been
rated in laboratory model studies (World Meteorological Organiza-tion,
1971). The transfer of a laboratory discharge rating to a struc-ture
in the field requires the existence, and maintenance, of
similitude between laboratory model and prototype, not only with
regard to the structure, but also with regard to the approach channel.
For example, scour and (or) fill in the approach channel will change
the head-discharge relation, as will algal growth on the control struc-ture.
Both the structure and the approach channel must be kept free
from accumulations of debris, sediment,, and vegetal growth. Flow
conditions downstream from the structure are significant only to the
extent that they control the tailwater elevation, which may influence
the operation of structures designed for free-flow conditions.
Because of the likelihood of the existence or development of condi-tions
that differ from those specified in a laboratory model study, the
policy of the Geological Survey is to calibrate the prototype control in
the field by discharge measurements for the entire range of stage that
is experienced. (See section in chapter 3 titled, “Artificial Controls.“)
In-place calibration is sometimes dispensed with where the artificial
control is a standard thin-plate weir having negligible velocity of
approach.
.I H I N-I’LA-I‘E WEIRS
The surface of the weir over which the water flows is the crest of the
weir. A thin-plate weir has its crest beveled to a chisel edge and is
always installed with the beveled face on the downstream side. The
crest of a thin-plate weir is highly susceptible to damage from floating
debris, and therefore such weirs are used as control structures almost
solely in canals whose flow is free of floating debris. Thin-plate weirs
are not satisfactory for use in canals carrying sediment-laden water
because they trap sediment and thereby cause the gage pool to fill
with sediment, sometimes to a level above the weir crest. The banks
of the canal must also be high enough to accommodate the increase in
stage (backwater) caused by the installation of the weir, the weir
plate being an impedance to flow in the canal. The commonly used
shapes for thin-plate weirs are rectangular, trapezoidal, and triangu-lar
or V-notch.

The information needed to compute the discharge over a thin-plate
weir is as follows:
1. Static head (h), which is the difference in elevation between
the weir crest and the water surface at the approach section;
the approach section is located upstream from the weir face
a distance equal to about 3h or more. (See section in chapter
2 titled, “Considerations in Specific Site Selection” for dis-cussion
of location of gage intakes.)
2. Length of crest of weir (b) if weir is rectangular or trapezoidal.
3. Width of channel in the plane of the weir face (B).
4. Angle of side slopes if weir is triangular or trapezoidal.
5. Average depth of streambed below elevation of weir crest (P).
P is measured in the approach section.
Flow over a rectangular thin-plate weir is illustrated in figure 143.
The discharge equation for this type of weir is:
Q = Cbh+, (56)
where
Q = discharge,
C = discharge coefficient,
b = length of weir crest normal to flow, and
h = static or piezometric head on a weir, referred to the weir
crest.
Information on discharge coefficients for rectangular thin-plate
weirs is available from the investigations of Kindsvater and Carter
(1959) and others, and is given in the previously cited WMO.Techni-cal
Note No. 117 (1971). Those investigations show that the coeffi-cient
for free discharge is a function of certain dimensionless ratios
which describe the geometry of the channel and the weir;
C = f(+ $J),, (57)
where E is the slope of the weir face; the other variables are depicted
in figure 143.
The relation between C, hlP and E for weirs with no side contrac-tion
CblB = 1.0) is shown in figure 144, where each of the four curves
corresponds to a particular value of E. The coefficient is defined in the
range of hlP from 0 to 5. The value of the coefficient becomes uncer-tain
at high values of hip. The greater the value of hlP, the greater
the velocity of approach, and therefore the greater the coefficient. The
coefficients in figure 144 are for use with English units, where all

linear measurements are expressed in feet and discharge is in cubic
feet per second. If linear measurements are expressed in meters and
discharge is in cubic meters per second, all values of C must be mul-tiplied
by the factor 0.552.
Side contractions reduce the effective length of the weir crest. That
effect is accounted for by multiplying the value of C from figure 144
by a correction factor that is a function of blB, hlP, and the degree of
rounding of the upstream vertical edge of the weir-notch abutments.
Rounding is a factor only in the situation where the horizontal weir
crest is set between vertical abutments. For a rectangular thin-plate
weir with sharp-edged entry, the correction factor is k,; appropriate
values of k, are obtained from the curves in figure 145. For a
FIGURE 143.-Definition sketch of a rectangular thin-plate weir.

4.6
2.0 4.0
h
3.0 5
L
P
FIGURE 144.-Discharge coefficients for full-width, vertical and inclined, rectangular
thin-plate weirs.
0.
-u’
0
0
FIGURE 145.-Definition of adjustment factor, k,, for contracted rectangular thin-plate
wews.

rectangular thin-plate weir having vertical abutments rounded with
radius r, the correction factor, when rlb>0.12, is assumed to equal (1
+ kJ2, where again k, is read from figure 145. For rounded abut-ments
having a lesser value of r/b, the correction factor is obtained by
interpolating between the appropriate k,. value from figure 145 and
the value (1 + kJ2. In other words, for given values of blB, hlP, and
rlb between 0 and 0.12, we interpolate linearly between the value of
k, (corresponds to r/b=O) and the value of (1 + kc)/2 (corresponds to r/b
= 0.12).
We will now prepare the rating for a hypothetical rectangular
thin-plate weir for the purpose of examining the implications of a
logarithmic plot of the rating. The discharges corresponding to var-ious
stages will be computed by use of the theoretical equation for a
rectangular weir, using figures 144 and 145 to obtain the constant in
the equation. We will assume that the computed discharges represent
the results of carefully made discharge measurements.
Assume that we have a rectangular thin-plate weir, with vertical
face, and that discharge measurements (computations) have been
made at heads ranging from 0.1 ft to 7.0 ft. The dimensions of the weir
using the symbols in figure 142 are given in table 16. The data and
computations are also shown in table 16 and should be self-explanatory.
The weir constant is actually equal to C(k,). The stages
and corresponding discharges are plotted on logarithmic graph paper
and fitted with a curve by eye in figure 146.
Figure 146 shows that a tangent can be fitted to the plotted points
at heads greater than 0.3 ft (G=1.3 ft). The intercept (PI of the tan-gent
at G - e= 1.0 ft is 67 ft3/s and the measured slope of the
tangent is 1.55. (Note that the slope of the rating curve Q/h is the
ratio of the horizontal distance to the vertical distance.) In accordance
with equation 53, the equation of the tangent is therefore Q =67h’.‘”
However, the equation for discharge over a rectangular weir is
Q=(Ck,b)h’.““. Therefore (Ck,b) must vary with stage, as we know it
does, and Ck,b=67h0+05; the exponent 0.05 is obtained by subtracting
the theoretical exponent 1.50 from the empirical exponent 1.55. Be-cause
b has a constant value of 20 ft, Ck,=3.35h”.“z; the coefficient
3.35 is obtained by dividing the original coefficient (67) by the value
of b (20 ft). We can extrapolate the tangent in figure 146 with some
confidence. If we wish to determine the discharge from the curve for a
gage height of 11 ft (h = 10 ft), the extrapolated value of Q is 2,380
ft”/s; that is, if a value of 10 ft is substituted in the equation
&=67h’.“‘, Q will equal 2,380 ft3/s. That value matches the
true value computed on the bottom line of table 16 for a gage height
of 11 ft.
Few experimental data are available for determining the discharge

Iiii iii i 1111II I II I I

coefficients of trapezoidal weirs (fig. 147). One exception is the verti-cal
Cippoletti weir, which is a sharp-crested trapezoidal weir whose
sides have a slope of 1 horizontal (x) to 4 vertical (y). The slope of the
sides is approximately that required to obtain a discharge through
the two triangular parts of the weir opening that equals the decrease
in discharge resulting from end contractions. In other words, the
Cippoletti weir acts as a rectangular thin-plate weir whose crest
length is equal to b and whose contraction coefficient, k,, is equal to
1.0. The dimension B (fig. 147) for a Cippoletti weir has little bearing
on the discharge. The equation used to compute discharge is again
Q =Cbh:p, (56)
and close approximations of values of C are obtained from figure 144.
The head, h, and height of notch, P, are both measured in the ap-proach
section.
If we compute the discharge for a vertical thin-plate Cippoletti weir
whose value of b is 20 ft and whose value of P is 2.0 ft, similar to the
dimensions used in computing the hypothetical rating shown in table
16, the rating will approximate that obtained for the thin-plate
rectangular weir of table 16. The only difference in discharge will be
that attributable to the fact that the value of k,. is 1.00 for all values of
head for the Cippoletti weir. A logarithmic plot of the rating (not
shown here) indicates that the equation for all but the very small
values of head is
&=69h’.“*, (English units)
meaning that C =3.45h”,“‘.
For trapezoidal weirs other than Cippoletti weirs, the general em-pirical
equation for discharge is
Q=Cb(h+h,.F, (58)
k B 4
FIGURE 147.-Sketch of upstream face of a trapezoidal weir.

, where h,., the velocity head at the approach section, is equal to V,%g, V,
being the mean velocity in the approach section and g being the
acceleration of gravity. The coefficient C and exponent N must be
determined from current-meter discharge measurements that cover
the entire range of stage that is experienced. If discharge meas-urements
are not available for the highest stage experienced, a rating
curve is obtained by plotting on logarithmic paper the head (G-e)
against discharge (Q) for the measurements that have been obtained,
and then fitting a curve to the plotted points. The upper end of that
curve should be a tangent, or possibly an extremely flat curve, that
can be extrapolated to the highest stage experienced. Because the
limiting shapes of a trapezoid are a rectangle at one extreme and a
triangle at the other, the slope of the tangent will lie somewhere
between 1.5, which is the theoretical slope for a rectangular weir, and
2.5 which is the theoretical slope for a triangular weir. The closer the
shape is to a rectangle, the closer the slope will be to 1.5; the closer
the shape is to a triangle, the closer the slope will be to 2.5.
The reader will note the difference in form between equations 56
and 58. Equation 56 uses static head (h), whereas equation 58 uses
total head (h + h,.). Velocity head is a factor in any discharge equation
for a weir. In the more modern laboratory studies of weir discharge,
the static-head term is used in the discharge equation, and velocity
head, as indicated by a term hlP, is used directly as a variable in the
determination of C. (See equation 57.) In older laboratory investiga-tions,
a more empirical approach for determining C was followed in
that the total-head term was used in the discharge equation and the
values of C that were determined do not vary directly with change in
velocity head. Both forms of the weir-discharge equations will be
found in this manual; the older type of equation is shown wherever it
has not been superseded by later laboratory studies.
TKIANGlJLAK OK V-NOTCH THIN-PLATE WEIR
Triangular or V-notch thin-plate weirs (fig. 148) are installed at
sites where low discharges occur; they are highly sensitive to low
flows but have less capacity than rectangular or trapezoidal weirs.
Because the area of the notch is invariably small compared to the
cross-sectional area of the channel, water is pooled upstream from the
weir and the approach velocity is necessarily low. The approach veloc-ity
head can usually be neglected in computing the discharge for a 90”
V-notch weir (8=90” in fig. 148). Actually, for values of 8 equal to or
less than go”, it has been specified that velocity of approach is negii-gible
if h/P is less than 0.4 and h/B is less than 0.2 (WMO Tech. Note
117, 1971). Whether or not the velocity head can be ignored in com-puting
discharges for V-notch weirs having central angles greater

304
COMPUTATION OF DISCHARGE
than 90” depends on the relative size of the areas occupied by the
water in the notch and the water in the approach section. Virtually no
experimental work has been done with triangular weirs having
significant velocity of approach, and all equations discussed below are
for installations where the velocity of approach can be neglected.
Furthermore the equations are applicable only for thin-plate V-notch
weirs whose faces are vertical.
Both the cross-sectional area of the flow in the notch and its veloc-ity
in the notch are functions of the head (h). Consequently, the gen-eral
equation for a triangular thin-plate weir is Q=Ch’ , and the con-stants
in that equation do not vary greatly from those in the following
equations:
&=2.5(tan W2)h’?,
where h is in feet and Q is in cubic feet per second; or
Q= 1.38(tan 8/2)hsiJ,
where h is in meters and Q is in cubic meters per second.
The head is measured in the approach section, a distance about 3h
upstream from the weir face. From an earlier discussion it is apparent
that the above equations will plot as straight lines on logarithmic
graph paper. The slope of the ratings will be 2.5, and the intercept,
where h=l, will be either 2.5 tan W2 or 1.38 tan W2, depending on
whether English or metric units are used.
V-notch weirs are most commonly built with a central angle of 90”.
Much experimental work has been done with thin-plate 90”
b El -I
P
FIGURE 148.-Sketch of upstream face of a triangular or V-notch weir.

V-notches, and the discharge equation usually recommended in the
U.S.A. is
Q =2.47h’,s’B (English units).
More precise values of the weir coefficient, which vary with h, are
given for use with metric units in WMO Technical Note No. 117.
The only other central angle that is commonly used in the U.S.A.
for V-notch weirs is 120”. The recommended discharge equation is
Q =4.35h’.,‘” (English units).
SUBMERGED THIN-PLATE WEIRS
Submergence occurs at a weir when the elevation of the
downstream water surface (tailwater) exceeds the elevation of the
weir crest (fig. 149). The tailwater elevation is measured downstream
from the turbulence that occurs in the immediate vicinity of the
downstream face of the weir. The degree of submergence is expressed
by the ratio h,lh. For any given head h, submergence has the effect of
reducing the discharge that would occur under the condition of free
flow; the greater the submergence ratio h,lh, the greater the reduction
in discharge. Villemonte (1947) combined the results of his tests with
those of several other investigators to produce the generalized rela-tion
shown in figure 150. Figure 150 is applicable to all shapes of
vertical thin-plate weirs. In that figure, the abscissa is the sub-mergence
ratio raised to a power N, where N is the exponent in the
free-flow discharge equation; for example, N=1.5 for a rectangular
weir and N=2.5 for a triangular weir. The ordinate in figure 150 is
the ratio of discharge under the submerged condition (Qs ) to free-flow
discharge (Q). The relation shown in figure 150 agrees reasonably
with the individual results obtained by the various investigators of
submerged-weir discharge. However, if great accuracy is essential, it
W.S.
FIGURE 149.--Sketch showing submergence of a weir.

is recommended that the particular weir be calibrated in the field or
in a laboratory under conditions similar to field conditions.
BROAD-CRESTED WEIRS
The term “broad-crested weir”, as used here, refers to any weir that
is not of the thin-plate type. The most common type of artificial con-trol
built in natural channels is the broad-crested weir. A structure of
that type has the necessary strength and durability to withstand
possible damage by floating debris. When installed in a stream chan-nel
that carries sediment-laden water, the weir is often built with a
gently sloping upstream apron (slope: 1 vertical to 5 horizontal) so
that there is no abrupt impedance to the flow and sediment is carried
over the weir and not deposited in the gage pool. Because the
1.0
0.9
08
a.;
Of
j-
Q, O.!
a
0.d
I-FIGURE
O.,
0.;
0.’
(
.BO.-Generalized relation of discharge ratio to submergence ratio for vertical
thin-plate weirs. (After Villemonte, 1947.)

backwater caused by a high weir can aggravate flood problems along
a stream, broad-crested weirs are usually built low to act as low-water
controls and they become submerged at intermediate and high
stages.
There are a myriad of crest shapes that can be used for broad-crested
weirs, and there will be no attempt to describe the charac-teristics
of the rating curve for each. Much of the material for such a
discussion can be found in WMO Technical Note No. 117 (1971), in a
report by Hulsing (1967), and in a textbook by King and Brater
(1963). Instead, this section of the report will present a discussion of
general principles as they apply to the definition of the discharge
rating and will present the approximate ratings for broad-crested
weirs commonly used as gaging-station controls in the U.S.A. The
weirs are all intended to be field calibrated by current-meter dis-charge
measurements.
Before proceeding to the discussion of broad-crested weirs com-monly
used in the U.S.A., it might be mentioned in passing that
perhaps the most popular weir for use as a gaging-station control in
Europe, and particularly in the United Kingdom, is the Crump weir
(World Meteorological Organization, 1971). The Crump weir is trian-gular
in cross section; the upstream face has a slope of 1 (vertical) to 2
(horizontal) and the downstream face has a slope of 1 (vertical) to 5
(horizontal). The crest, or apex of the triangular cross section, is
usually horizontal over its entire length (b), but for greater sensitiv-ity
the crest may be given the shape of a flat Vee, the sides of which
often have a slope of 1 (vertical) to 10 (horizontal). The basic equation
for the Crump weir with horizontal crest is,
Q = Cb (h +h,.j3P,
where C equals about 3.55 when English units are used and 1.96
when metric units are used.
FLAT-CRESTED RECTANCL’LAR VEIR
The simplest type of broad-crested weir is one that is rectangular in
cross section and whose crest is horizontal over its entire length, b.
The basic discharge equation for that weir is Q =Cb(h + h,.)‘.j, where h,
is the head attributable to velocity of approach. The coefficient C will
increase with stage in the manner shown in figure 151, and h,. will
also increase with stage as a result of the velocity of approach increas-ing
with stage. (Figure 151 also shows the relation of C to stage for
flat-crested weirs with sloping faces.) The rating curve for a flat-crested
rectangular weir, when plotted on logarithmic graph paper,
will be a straight line except for extremely low stages. The equation

308
of the line will be of the form Q =p(G -e)‘, where the slope of the line,
N, will have a value greater than 1.5 because both the weir coefficient
and the velocity head increase with stage.
Most flat-crested rectangular weirs are not sufficiently sensitive at
low flows. To increase the low-flow sensitivity, the crest is often
modified as shown in figure 152. Instead of the crest being horizontal
over its entire length, b, the crest is given a gentle slope from one
streambank to the other, or the crest is given the shape of an ex-tremely
flat Vee or catenary. As a result of this modification, the area
of flow over the weir is triangular, or nearly so, at low flows and
approximately rectangular at high flows. In other words, the length of
weir crest that is utilized by the flow varies with stage until the stage
rises high enough to flow over the entire length of the crest (b in fig.
152). In the general equation for the weir discharge, Q =Cb(h +h, )1.5,
not only do C and h, increase with stage, but length of weir crest, b,
also increases with stage, as stated in the preceding sentence. Con-sequently
if the weir rating plots as a straight line on logarithmic
graph paper in accordance with equation, Q=p(G-eY,the slope of the
line, N, will be considerably greater than 1.5, and invariably will be
greater than 2.0.
3.80
t i i i i i i i “h i i i
3.20
3.00
0
2.80
FIGURE 151.-Coefficients of discharge for full-width, broad-crested weirs with
downstream slope G 1:l and various upstream slopes. (Slope is the ratio of horizon-tal
to vertical distance.)

NO’I‘~.HEI~ FLAT-C:KESI‘ED RECTAKGULAR VEIR
Figure 153 shows the notched flat-crested rectangular weir that is
the control for a gaging station on Great Trough Creek near Markles-burg,
Pa.
Because there is a sharp break in the cross section at gage height
1.4 ft, a break occurs in the slope of the rating curve at that stage. The
gage-height of zero flow for stages between 0.0 and 1.4 ft is 0.0 ft; for
stages above 1.4 ft, the effective zero flow is at some gage height
between 0.0 and 1.4 ft. If the low end of the rating is made a tangent,
the gage height of zero flow (e) is 0.0 ft, and the slope of this tangent
turns out to be 2.5, which, as now expected, is greater than the
theoretical slope of 1.5. The upper part of this rating curve is concave
upward because the value of e used (0.0 ft) is lower than the effective
value of zero flow for high stages.
If the upper end of the rating is made a tangent, it is found that the
value of e, or effective zero flow, must be increased to 0.6 foot. Because
we have raised the value of e, the low-water end of the curve will be
concave downward. The high-water tangent of the curve, principally
because of increased rate of change of velocity of approach, will have a
- -- _w.s.2 ------------- V ------ -----J&s. ---1-- -- V
h,T h2
A
b -I
--v- _____ f
hz
B
FIGURE 152.-Sketch of upstream face of flat-crested weir with (A) sloping crest and
(B) catenary crest.

slope that is greater than that of the low-water tangent of the curve
previously described; its slope is found to have a value of 3.0.
The low-water tangent for the notched control, which is defined by
discharge measurements, warrants further discussion. Its slope of 2.5
is higher than one would normally expect for a simple flat-crested
rectangular notch. One reason for the steep slope is the fact that the
range of stage involved, 0.0 ft to 1.4 ft, is one in whkh the theoretical
weir coefficient C increases very rapidly with stage. A more impor-tant
reason is the geometric complexity of the notch which is not
indicated in figure 153. At the downstream edge of the notch is a
sharp-edged plate; its elevation is at 0.0 ft, but the sharp edge is about
0.1 ft higher than the concrete base of the notch. The details of the
notch are not important to this discussion; they are mentioned here
only to warn the reader not to expect a slope as great as 2.5 in the
rating for a simple flat-crested rectangular notch. In fact, the sole
purpose here of discussing the low-water tangent of the rating curve
is to demonstrate the effect exerted on the curve by varying the
applied values of e. The low-water end of a rating curve is usually
well defined by discharge measurements, and if it is necessary to
DISCHARGEI,N CUBICF EETP ER SECOND
FIGURE 153.-Rating curve for a notched broad-crested control at Great Trough Creek
near Marklesburg, Pa.

extrapolate the rating downward, it is best done by replotting the
low-water end of the curve on rectangular-coordinate graph paper,
and extrapolating the curve down to the point of zero discharge. (See
section titled, “Low-Flow Extrapolation.“)
X’KENI‘ON-1‘YPE CONTROL.
The so-called Trenton-type control is a concrete weir that is fre-quently
used in the U.S.A. The dimensions of the cross section of the
crest are shown in figure 154. The crest may be constructed so as to be
horizontal for its entire length across the stream, or for increased
low-flow sensitivity the crest may be given the shape of an extremely
flat Vee. For a horizontal crest, the equation of the stage-discharge
relation, as obtained from a logarithmic plot of the discharge meas-urements,
is commonly on the order of Q=3.5bh’~fis(English units).
The precise values of the constants will vary with the height of the
weir above the streambed, because that height affects the velocity of
approach. The constants of the equation are greater than those for a
flat-crested rectangular weir (see section titled, “Flat-crested
Rectangular Weir”) because the cross-sectional shape of the Trenton-type
control is more efficient than a rectangle with regard to the flow
of water.
When the Trenton-type control is built with its crest in the shape of
a fiat Vee, the exponent of h in the discharge equation is usually 2.5
or more, as expected for a triangular notch where velocity of approach
FIGURE 154.-Cross se&on of Trenton-type control

is significant. Again, the precise values of the constants in the dis-charge
equation are dependent on the geometry of the installation.
COLUMBUS-1‘YPE CONTKOL
One of the most widely used controls in the U.S.A. is the
Columbus-type control. This control is a concrete weir with a
parabolic notch that is designed to give accurate measurement of a
wide range of flows (fig. 155). The notch accommodates low flows; the
main section, whose crest has a flat upward slope away from the
notch, accommodates higher flows. The throat of the notch is convex
along the axis of flow to permit the passage of debris. For stages above
a head of 0.7 ft, which is the elevation of the top of the notch, the
elevation of effective zero flow is 0.2 ft, and the equation of discharge
is approximately,
Q =8.5(h -0.2)“.” (English units)
The precise values of the constants in the equation will vary with
conditions for each installation. The shape of the crest above a stage
of 0.7 ft is essentially a flat Vee for which the theoretical exponent of
head is 2.5 in the discharge equation. However, the actual value of
the exponent is greater than 2.5 principally because of the increase of
velocity of approach with stage.
SUHMEKGED KKOAD-CKESI‘ED :EIKS
Weir submergence was defined earlier in the section titled, “Sub-merged
Thin-Plate Weirs.” As in the case of thin-plate weirs, for a
given static head (h) the discharge decreases as the submergence
ratio (h,/h) increases. Little quantitative data are available to define
the relation of discharge ratio to submergence ratio for the many
types of broad-crested weir. However, it is known that for horizontal
crests the submergence ratio must be appreciable before any
significant reduction in discharge occurs. This threshold value of the
submergence ratio at which the discharge is first affected ranges from
about 0.65 to 0.85, depending on the cross-sectional shape of the weir
crest.
Flumes commonly utilize a contraction in channel width and free
fall or a steepening of bed slope to produce critical or supercritical
flow in the throat of the flume. The relation between stage measured
at some standard cross section and discharge is thus a function only of
the characteristics of the flume and can be determined, on an interim
basis at least, prior to installation.
In the section in chapter 3 titled, “Artificial Controls,” it was men-tioned
that flumes may be categorized with respect to the flow regime

that principally controls the measured stage; that is, a flume may be
classed as either a critical-flow flume or a supercritical-flow flume.
The most commonly used critical-flow flume is the Parshall flume,
and it is the only one of that type that will be described here. The
Profile of weir crest and notch
Coordinates of notch profile, in feet
XN YN
7 Coordinates of cross section
-x
Cross sectlon of weir
0.00 0.00
,046 .I
,108 .2
,192 .3
,302 .4
,452 .5
,665 6
1 .oo 7
‘es:1, in feet
of weir cr
X
0.0
.l
.i
.4
.6
8
Y
0.126
,036
,007
.ooo
,007
,060
,142
1.0 ,257
1.2 ,397
1.4 565
1.7 ,870
2.0 1.22
25 1.96
FIGURE 155.-Dimensions of Columbus-type control.

supercritical-flow flume is less widely used, but fills a definite need.
(See section in chapter 3 titled “Choice of an Artificial Control.“) Of
that type of flume, the trapezoidal supercritical-flow is preferred by
the Geological Survey; it too will be described here.
PARSHALL FLUME
The principal feature of the Par-shall flume is an approach reach
having converging sidewalls and a level floor, the downstream end of
which is a critical-depth cross section. Critical flow is established in
the vicinity of that cross section by having a sharp downward break
in the bed slope of the flume. In other words, the bed slope
downstream from the level approach section is supercritical. The
primary stage measurement is made in the approach reach at some
standard distance upstream from the critical-depth cross section.
The general design of the Parshall flume is shown in figure 156.
Table 17 gives the dimensions corresponding to the letters in figure
156 for various sizes of flumes. The flumes are designated by the
width (W) of the throat. Flumes having throat widths from 3 in. to 8 ft
have a rounded entrance whose floor slope is 25 percent. The smaller
/-
Note: Three-Inch to eight-foot flumes hove
rounded opprooch wmgwolls
PLAN VIEW
SIDE VEW
FIGURE 156.-Configuration and descriptive nomenclature for Parshall flumes.

SIMPLE STAGE-DISCHARGE RELATIONS

and larger flumes do not have that feature, but it is doubtful whether
the performance of any of the flumes is significantly affected by the
presence or absence of the entrance feature as long as approach condi-tions
are satisfactory.
Parshall flumes have provision for stage measurements both in the
approach reach and in the throat reach, but the downstream gage is
required only when submerged-flow conditions exist. The datum for
both gages is the level floor in the approach. The raised floor, length G
in figure 156, in the downstream diverging reach is designed to re-duce
scour downstream and to produce more consistent stage-discharge
relations under conditions of submergence. The percentage
of submergence for Parshall flumes is computed by the formula,
+ 100.
A
Where free-flow conditions exist for all flows, the downstream gage,
hB, may be omitted and the entire diverging reach may be dispensed
with if desired. That simplification has been used in the design of
small portable Parshall measuring flumes. (See section in chapter 8
titled, “Portable Parshall Flume.“)
Tables 18 and 19 summarize the relation of discharge to stage at h,
under conditions of free flow (low stage at h,) for flumes of the various
sizes. Although the free-flow stage-discharge relations for the various
flumes were derived experimentally, all relations can be expressed
closely by the following equation, (Davis, 1963),
QII’
y” + 2Y,,” (1 + 0.4X,,)”
= 1.351 Q,, “li’j (English units)
in which
Y,, = nondimensional depth, y,lb
Q,, = nondimensional discharge, Qlg’“b5p
X,, = nondimensional distance, xlb
y1 = depth at measuring section
b = channel width at throat
Q = discharge
g = acceleration of gravity
x = distance from throat crest to measuring section.
For flumes with throat widths no greater than 6 ft, the following
simplified form of the above equation (Dodge, 1963) can be used:
y,, = l.lgQ,,” 64.7 X,, U.O,!,, (60)

When the stage h, is relatively high, the free-flow discharge cor-responding
to any given value of h,, is reduced. The percentage of
submergence, or value of [ (hJh,)x 1001, at which the free-flow
discharge is first affected, varies with the size of flume. For flumes
whose throat width is less than 1 ft, the submergence must exceed 50
percent before there is any backwater effect; for flumes with throat
width from 1 to 8 ft, the threshold submergence is 70 percent; for
flumes with throat width greater than 10 ft, the threshold sub-mergence
is 80 percent. Figure 157 shows the discharge ratings for
Parshall flumes, from 2 inches to 9 inches, under both free-flow and
submergence conditions. Figure 158 shows the correction in dis-charge,
which is always negative, that is to be applied to free-flow
discharges for various percentages of submergence and various val-ues
of h , , for flumes having throat widths between 1 and 50 feet. The
appropriate correction factor (&) for flume size is applied to the cor-rections
read from the graphs. In other words,
where
Q,, = discharge under submergence conditions,
Q, = discharge under free-flow conditions, and
Q,. = discharge correction unadjusted for flume size.
The stage-discharge relations, both for free-flow and submergence
conditions, given in the preceding tables and graphs, should be used
only as guides or as preliminary ratings for Parshall flumes built in
the field. Those installations should be field-calibrated because the
structural differences that invariably occur between model and pro-totype
flume usually cause the discharge rating for the field structure
to differ from the experimental ratings given in this manual.
TABLE 18.-Discharge table for Parshall flumes, sizes 2 inches to 9 inches, for free-flow
conditions
[Discharges far standard 3.Inch Parshall flumes are slightly less than those for the modified 3.Inch Parshall flume
drscussed I” chapter 8: see table 14 1

FIG
.5
0 IO
0
.?
Percent sut ,mer
hg/hA x 100
3 IN. FLUME /fhk%+
0.1 0.2 0.4 0 6 0.8 IO 20 4.0 6 0 8.0 IO
9lN. FLUME DISCHARGE, IN CUBIC FEET PER SECOND
:URE 157.-Discharge ratings for “inch” Parshall flumes for both free-flow and sub-mergence
conditions.

320
COMPUTATIqN OF DISCHARGE
TRAPE%OIDAL S~PER<:RITI<:AI,-FI,(~~~ FLL’hfE
The principal feature of the trapezoidal supercritical-flow flume is a
reach of flume (throat) whose bed has supercritical slope, upstream
from which is a critical-depth cross section. The general design of the
flume and the dimensions for the flumes of three throat widths that
have been installed by the Geological Survey are shown in figure 159.
The purpose of having the flume trapezoidal in cross section is to
increase the sensitivity of the stage-discharge relation, particularly
at low flows. Wide latitude exists with regard to the height (E) of the
z 1
.
2 I
2 0
g 0
= 0
W
k 0
3
0
DISCHARGE CORRECTION, Qc , IN CUBIC FEET PER SECOND
a. FLUMES I-B ft.
60
50
6
5
56 810 2 4 6810 20 40 60 80 100 200 4oc
DISCHARGE CORRECTION, 0, , IN CUBIC FEET PER SECOND
b FLUMES IO- 50 ft
FIGURE 158.-Correction factors for submerged flow through l- to 50-ft Parshall
flumes.

sidewalls that can be used, and thus the range of discharge that can
be accommodated by a supercritical-flow flume of any particular
throat width is quite flexible. Stage (vertical depth of flow) is meas-ured
at a cross section at midlength of the throat reach, gage datum
being the floor of the flume at the stage-measurement site. The meas-urement
of stage must be precise because the stage-discharge relation
for super-critical flow is extremely insensitive-a small change in
stage corresponds to a large change in discharge.
Were it not for the severe width constriction at the downstream end
of the converging reach, critical flow would occur at the break in floor
slope at the downstream end of the approach reach and flow would be
super-critical at all cross sections downstream from the approach
reach. However for all but extremely low flows, the sharp constriction
in width resulting from the use of a convergence angle ($1 of 21.8
(fig.1591 causes backwater that extends upstream into the approach
Note-Height of wall E) is depenoent on magnitude of maxlmum discharge to be gaged
/ Stage-measurement
>
site
FIGURE X9.-Configuration and dimensions of trapezoidal supercritical-flow flumes of
three throat widths.

322
reach. As a result critical depth occurs at the most constricted cross
section in the converging reach, the flow being subcritical in the
approach and converging reaches and supercritical in the throat
reach. That is seen in figure 5 (chap. 3) which is a photograph of a
3-foot trapezoidal flume in Owl Creek in Wyoming. The purpose of the
converging reach is to obtain an increased velocity at the critical-depth
cross section and thereby reduce the likelihood of debris deposi-tion
at that cross section; such deposition could affect the stage-discharge
relation in the throat of the flume.
The measured stage corresponding to any discharge is a function of
the stage of critical depth at the head of the throat reach and the
geometry of the throat reach upstream from the stage-measurement
cross section. Consequently a theoretical rating for all but the small-est
discharges can be computed by use of the Bernoulli or total-energy
equation for the length of throat reach upstream from the
stage-measurement site (fig. 160). By equating total energy at the
critical-depth cross section (c) at the head of the throat reach to total
energy at the stage-measurement cross section (m), we have,
E2+h, +z, ==
2g 29
+ h,,, + z, + h,, (61)
where
V is mean velocity,
g is acceleration of gravity,
h is vertical depth,
z is elevation of flume floor above any arbitrary datum plane,
and
hf is friction loss.
We make the assumption that the friction loss hf in the short reach
is negligible and may be ignored. Then by substituting, in equation
61 values from the two equations
Q = A,V,. = A,,,V,,, and AZ =Z, -Z,,,,
we obtain
Q’ Q’ - +h,+Az=---
2gA’, %A 2,,,
+ h,,, (62)
From the properties of critical-depth flow (Chow, 1959, p. 641, the
critical-section factor (J) is computed by the formula
(63)

where A, is the area and T, is the top width at the critical-depth cross
section. The discharge (Q) at the critical-depth cross section is
Q=JG
By assuming a depth (h,.) at the critical-depth cross section, we can
compute Q and A,. , and thus the values of all terms on the left side of
equation 62 will be known for any chosen value of h,. Because h,,, is
uniquely related to A,,, , equation 62 can be solved by trial and error to
obtain the depth (stage) at the measurement cross section correspond-ing
to the value of Q that was computed earlier.
The entire procedure is repeated for other selected values of h, to
provide a discharge rating curve for the entire range of discharge.
The value of h,. corresponding to the maximum discharge to be gaged
represents the height to which the sidewalls of the throat section
must be built to contain that discharge. An additional height of at
least 0.5 ft should be added for freeboard to accommodate surge and
wave action.
The computed discharge rating should be used only until the rating
can be checked by current-meter discharge measurements. The
sources of error in the computed rating are uncertainty as to the exact
location of the critical-depth cross section for any given discharge and
neglect of the small friction loss (h,). However, the general shape of
the discharge rating curve will have been defined by the computed
values, and relatively few discharge measurements should be re-quired
for any needed modification of the rating.
k Approach reach JIc Converging reach +-Throat reach----(
FIGURE 160.~-Sketch illustrating use of the total-energy (Bernoulli) equation.

324
The total-energy equation (eq. 61) should also be applied to the
converging reach to obtain the required height of the sidewalls at
the upper end of the converging reach. The height plus an addi-tional
freeboard height of at least 0.5 ft should be used for the
sidewalls in the approach reach. In applying equation 61 to the con-verging
reach, the value of discharge used is the maximum dis-charge
that is to be gaged, and the depth used at the lower end
of the converging reach is the corresponding critical depth (h,) that
was computed earlier for the throat reach.
The solid-line curves on figures 161-163 are the theoretical dis-charge
rating curves for the flumes of the three throat widths that
have been field tested. The agreement between measured and
theoretical discharges has generally been good except at extremely
low stages. Nevertheless the theoretical curves should be considered
as interim rating curves for newly built flumes until later meas-urements
either corroborate the ratings or show the need for
modification of the ratings. It is expected that the stage-discharge
relation will not be affected by submergence, as long as submergence
percentages do not exceed 80 percent. (Percentage of submergence for
a given discharge is defined as the ratio, expressed as a percentage, of
the stage in the natural channel immediately downstream from the
throat reach to the stage at the stage-measurement site, both stages
being referred to the floor elevation of the flume at the stage-measurement
site.)
5
4
3
0.5 0.7 1 2 3 456 810 20 30 40 60 80100 200 300
DISCHARGEI,N CUBIC FEET PER SECOND
FIGURE 161.--Stage-discharge relation and significant depth-discharge relations for
l-ft trapezoidal supercritical-flow flume.

Measurement and Computation of Streamflow: Volume 2 Computation of Discharge Paper 2175 1982

Measurement and Computation of Streamflow: Volume 2 Computation of Discharge Paper 2175 1982

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Measurement and Computation of Streamflow: Volume 2 Computation of Discharge Paper 2175 1982